The midfielder was substituted in the 29th minute after picking up the injury in a drop-ball situation with the Cobblers' John-Joe O'Toole.
A foul was given for the challenge but the Northampton midfielder escaped punishment.
Pompey have confirmed that Rose has been taken to hospital for an x-ray.
He was replaced by Adam May in the centre of the Blues midfield.
Rose has made 19 appearances for Pompey this season.
Goals from Matty Kennedy and OlI Hawkins (2) earned Kenny Jackett's side victory.